Unless you’re going to outlaw cabin-size bags entirely, no narrowbody aircraft, save maybe the C500, has enough room for everyone to put even a single bag in the bins. If your bag is small enough to fit under the seat in front of you, that’s where it belongs. If your only concern is that you don’t want to pay to check it, they’ll be happy to check it at the gate for no charge. Some airlines will even allow you to board early if you do this, although (a) not UA and (b) I don’t know why you’d want to board early if you don’t have a bag that needs to go into the overhead.

If, after everyone has boarded, there’s room for your small bag in the overhead, go for it.

The idea that each passenger has some sort of dedicated overhead bin allotment is pure fantasy. They’re shared space. In fact, it is demonstrably untrue that each passenger has a “right” to put a bag in the overhead, because if each passenger tries to exercise that right, the bins won’t close.